Spectral concentration and perturbed discrete spectra

AbstractWe examine spectral concentration for a class of Sturm-Liouville problems on [0, ∞), a typical example being y″(x) + (λ − x + cx2)y(x) = 0. The discrete spectrum for c = 0 leads to spectral concentration in the continuous spectrum for c > 0, and we use a new formula for the spectral function to make a detailed computational investigation of the way in which spectral concentration occurs. In particular, we find that, as c decreases, spectral concentration arises first from the lowest unperturbed eigenvalue and then in turn from these eigenvalues in increasing order of size